COMPATIBILITY OF EXTENSIONS OF A COMBINATORIAL GEOMETRY

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Abstract

Two extensions of a geometry are compatible with
each other if they have a common extension. If the given
extensions are elementary, their compatibility can be
intrinsically described in terms of their corresponding
linear subclasses. Certain adjointness relation between
an extension of a geometry and the geometry itself is
also discussed.
Any extension of a geometry G by a geometry F determines
and is determined by a unique quotient bundle on G
indexed by F. As a study of the compatibility among
given quotients of a geometry, we look at the possibility
of completing to F-bundles a family of quotients indexed
by a set I of flats of F. If the indexing geometry F is
free and if the set I is a Boolean subalgebra or a sublattice
of F, for any family Q(I) of quotients of a geometry
G, there is a canonical construction which determines
its completability and at the same time produces
the extremal completion if it is a partial bundle.
Geometries studied in this dissertation are furnished
with the weak order. Almost invariably, the Higgs' lift
construction, in a somewhat generalized sense, constitutes
a convenient and indispensable means in various of the
extremal constructions.